Now, we present the formula for the fundamental function of \(\mathcal {M}_{\varphi ,w}\)  with respect to any extended real valued Orlicz functions such that they can assume as their values 0 or \(\infty \)  on \((0, \infty )\) . The similar formula with respect to a finite, strictly increasing Orlicz function has been considered in [1]}. However, for studying certain geometrical properties of an Orlicz-Lorentz space \(\Lambda _{\varphi ,w}\)  through its Köthe dual \(\mathcal {M}_{\varphi _*,w}\) , we may encounter the cases where \(\varphi _*(v) = 0\)  or \(\varphi _*(v) = \infty \)  for some \(v \in (0, \infty )\) . For instance, if \(\varphi (u) = ku\)  for some \(k > 0\) , then \(\varphi _*(v) = 0\)  on the interval \([0, k]\)  and \(\varphi _*(v) = \infty \)  on \((k, \infty )\) .
Therefore, for the space \(\mathcal {M}_{\varphi ,w}\)  we consider \(\varphi :\mathbb {R}^{+} \rightarrow [0, \infty ]\)  that is a convex, left-continuous function where \(\varphi (0) = 0\)  and \(\varphi \)  is not identical to the zero function or the infinity function on \(\mathbb {R}^{+}\) . Here as well as the constants \(a_{\varphi }\)  and \(d_{\varphi }\) , we will also consider the constant \(b_{\varphi }\)  given by
\(b_{\varphi } = \sup \lbrace u \ge 0: \varphi (u)< \infty \rbrace .\) 
